In this section we will sketch an alternative proof that \(Gr_{iso}(p,2p+1)\)  and the connected components of \(Gr_{iso}(p,2p)\) , in their Plücker embedding, are defined by linear equations and quadrics of rank at most 4, using the Cartan embedding (sometimes called spinor embedding), cf. [1]}.
The proof follows by combining the following facts:
